Scaling limits of discrete snakes with stable branching

Cyril Marzouk (1, 2)

1 LM-Orsay - Laboratoire de Mathématiques d'Orsay
2 CMAP - Centre de Mathématiques Appliquées - Ecole Polytechnique

We consider so-called discrete snakes obtained from size-conditioned critical Bienaym\'e-Galton-Watson trees by assigning to each node a random spatial position in such a way that the increments along each edge are i.i.d. When the offspring distribution belongs to the domain of attraction of a stable law with index $\alpha \in (1,2]$, we give a necessary and sufficient condition on the tail distribution of the spatial increments for this spatial tree to converge, in a functional sense, towards the Brownian snake driven by the $\alpha$-stable L\'evy tree. We also study the case of heavier tails, and apply our result to study the number of inversions of a uniformly random permutation indexed by the tree.

Domaines: Probabilités [math.PR]

Cyril Marzouk. Scaling limits of discrete snakes with stable branching. Annales de l'Institut Henri Poincaré (B) Probabilités et Statistiques, 2020, 56 (1), pp.502-523. ⟨10.1214/19-AIHP970⟩. ⟨hal-01718881⟩